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I was given this statement:

where $W\in\mathbb R^{4\times 10}$ and $\vec{b}\in\mathbb R^4$.

Now, I understand that $ℝ^{4}$ means that the set of real numbers in four dimensions. However, I am not sure what $ℝ^{4\times10}$ mean?

Pedro
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Pablo
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1 Answers1

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The notation

$$ \mathbb{R}^{m \times n} $$

is sometimes used to refer to the vector space of two-dimensional arrays of real numbers consisting of $m$ rows and $n$ columns.

It is rather common to identify this with the space of $m \times n$ real-valued matrices.

There is a nice coincidence of notation, since as abstract vector spaces there is an isomorphism

$$ \mathbb{R}^{m \times n} \cong \mathbb{R}^{mn} $$

obtained by rearranging the entries into a one-dimensional array.

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    Regarding the second sentence, do you not define an $m \times n$ real-valued matrix to be a two dimensional array of real numbers consisting of $m$ rows and $n$ columns? – littleO Jun 07 '18 at 01:45
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    Was the distinction between two dimensional array and matrix really necessary/helpful? –  Jun 07 '18 at 01:46
  • I didn't mean for that to come of snarky, sorry. It's an actual question, lol –  Jun 07 '18 at 01:47
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    @littleO: I consider the term "matrix" to imply a context with additional algebraic structure, namely matrix multiplication. –  Jun 07 '18 at 01:56